Sharp estimates for a class of hyperbolic pseudo-differential equations

Transcription

1 Results in Math., 41 (2002), Sharp estimates for a class of hyperbolic pseudo-differential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic pseudo-differential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L p L q, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R 1+n with n 4 (total dimension 5), we give a complete list of L p L q properties. In particular, this contains the very important case R Introduction The solutions of the Cauchy problem for the hyperbolic partial differential and pseudo-differential operators have been under study for a very long time. We will consider the situation when the j-th Cauchy data is in the Sobolev spaces L p m j, and we will study the question for which m j the fixed time solutions of the Cauchy problem belong to L p. Using standard methods one gets estimates for solutions in more general Sobolev spaces L p α, as well as Lipschitz and other function spaces. For p = 2, L 2 estimates correspond to the conservation of energy law and are relatively easy to obtain. However, for p 2, the problem becomes more subtle even for the wave equation. Some earlier estimates for the wave equation with variable coefficients in L p spaces can be found in [1], [5], [6], [7], [18], [20]. The case of compact Riemannian manifold is treated in [2]. The general approach to the L p estimates for solutions of hyperbolic Cauchy problems is to write them as a sum of time dependent Fourier integral operators applied to Cauchy data. This method is described in [3], [4] for differential operators, and in [24] for operators differential in time, but pseudo-differential in the space variables. In this way the estimates reduce to the corresponding L p estimates for time dependent Fourier integral operators. General regularity properties of Fourier 0 Mathematics Subject Classification (1991): 35A20, 35S30, 58G15, 32D20. 1

2 integral operators in L p and other function spaces can be found in [17], [19], [8], [13]. General L p properties were established in [16] and applied to solutions of strictly hyperbolic Cauchy problem. However, in a number of cases these results can be improved, see, for example, [9] and [13]. Regularity results for Fourier integral operators with complex phases and estimates for solutions of non-hyperbolic problems will appear in [14]. In this paper we will consider a class of constant coefficient operators, however allowing the dependence on time. In particular, our class includes constant coefficient differential or pseudo-differential operators. The L p L p estimates for hyperbolic operators with constant coefficients were treated in [21], [22] for convex and nonconvex characteristics, respectively. There the estimates for n = 2, 1 p 2, and n 3, p = 1, 2, were obtained. Several problems for constant coefficient operators in a 3 dimensional case were considered in [23]. In this paper we will allow operators to be differential in time, but pseudodifferential in space. In order to simplify the notation we single out one of the variables and call it time t. In particular, in R 1+n, this allows us to use the L p estimates for the Fourier integral operators in R n. Let X be a compact n dimensional manifold and let P (t, t, x ) = m t + m P j (t, x ) m j t (1) be a strictly hyperbolic differential pseudo-differential operator of order m, with t R and x X. We assume that the coefficients P j are classical symbols of order j depending smoothly on t. We also assume that P j are translation invariant in the sense that they do not depend on the x variable. The Cauchy problem for P is the equation { P u(t, x) = 0, t 0, j (2) t u t=0 = f j (x), 0 j m 1. As usual, D t = i t and D xj = i xj. Let σ P (t, τ, ξ) be the principal symbol of P in (2), that is the top order part of the symbol of P (t, D t, D x ), homogeneous of degree m. When operator P is strictly hyperbolic, problem (2) is well posed and its solution operator is given by a sum of elliptic Fourier integral operators ([24]). The strict hyperbolicity means that the principal symbol σ P (t, τ, ξ) has m real distinct roots in τ, τ j (t, ξ), which are real, homogeneous of degree one in ξ and smooth in t. Note that P (t, τ, ξ) is a polynomial in τ. It follows that the principal symbol σ P of P can be decomposed in the product m σ P (t, τ, ξ) = (τ τ j (t, ξ)). In this paper we will assume that σ P is real analytic in ξ, which is, for example, the case for differential operators P. The result of [16] states that if f j L p α+(n 1) 1/p 1/2 j (X), 0 j m 1, it follows that the solution u(t, ) L p α(x). Moreover, these orders are sharp when for 2

3 every t in the complement of a discrete set in R, at least one of the roots τ j is elliptic in ξ. However, in many cases this condition is not satisfied. It turns out that L p properties of solutions depend on the maximal rank k of ξξ 2 τ j(t, ξ), for j = 1,..., m. As usual, it is difficult to construct explicit examples of strictly hyperbolic equations with specific properties. However, one can readily see that there are many microlocal examples. Let us mention one specific example which still presents an open problem. Let the principal symbol σ P (τ, ξ) of the constant coefficients operator P in R R 5 be given by ( ) ( ) τ ξ1 ξ2ξ (ξ 3 ξ 5 ξ 2 ξ 4 ) 2 ξ5 3 τ + ξ1 ξ2ξ (ξ 3 ξ 5 ξ 2 ξ 4 ) 2 ξ5 3, microlocally in a small conic neighborhood C of ξ = (1, 1, 1, 1, 1). In this paper we will present sharp L p estimates for operators in R R n with n 4. However, in this example n = 5, and it is an open question whether in this case the loss of regularity in L p spaces is α p = 3 1/p 1/2 (since in this case k = 3), if we consider the Cauchy data f j with their wave fronts contained in C. This corresponds to a similar question for Fourier integral operators ([15]). We will give the sharp estimates for the class (1) in L p and Lipschitz spaces. By complex interpolation we also get sharp L q estimates for the solution when f j are in Sobolev L p spaces with q p. In the most important case of R 1+3 and in R 1+n, n 4, a complete list of sharp estimates is provided by Theorem 2.1. Dependent of the roots τ j, there is a certain loss of smoothness, regulated by α p of Theorem 2.2. We use the analysis of [13] and [9] to deduce the regularity results without any ellipticity assumption on τ j. This is based on the study of the factorization condition for Fourier integral operators ([13], [9]). We will make a technical assumption that X is real analytic, which is not restrictive since we are mainly interested in subsets of R n. A peculiar situation occurs when there is no loss of smoothness in the problem, that is when the problem resembles an elliptic problem with the solution given by a pseudo-differential operator. Technically this means that, f j L p m j, 0 j m 1, imply u(t, ) L p m. This is for example the case when p = 2. If p 2, this is still possible, but the operator P must assume a rather special form. This is discussed in Theorem 2.4. For our argument we use the representation formula established in Lemma 3.1. For an underlying relation to the Fourier integral operators and their singularities we refer to [13] for the case of differential equations and to [14] for the case of a non-hyperbolic class of equations. In a way, the present paper can be regarded as a supplement to [9], [10], providing proofs and extending results announced in [10]. Let us also note that due to editorial reasons the general perspective for results of this paper can be found in [13] and in [14]. The present paper is based on the preprint [12]. I would like to express my gratitude to professor Duistermaat for the discussions on Fourier integral operators and to professor Sogge for the discussions we had during my one year visit to the Johns Hopkins University. I am also grateful to the referee for a number of valuable remarks. 3

4 2 Main Results Let P, σ P (t, τ, ξ) and τ j be as in the introduction. The solution of problem (2) for small t can be obtained in the following way. Let Φ j (t, x, ξ) be the solution of the following eikonal equation: { Φ t j(t, x, ξ) = τ j (t, x Φ j ), (3) Φ j (t, x, ξ) t=0 = x, ξ. Then the solution u(t, x) can be written as a smooth function plus a finite sum of t-dependent elliptic Fourier integral operators u(t, x) = m l=0 R n R n e 2πi(Φ j(t,x,ξ) y,ξ ) a jl (t, x, ξ)f l (y)dξdy, (4) with canonical relations locally defined by the solutions Φ j of (3) and symbols a jl S l, satisfying the transport equations, see [24]. The expression (4) is smooth in t. For a fixed t, the canonical relation of the solution operator in (4), is equal to C t = m {(x, ξ, y, η) : χ t,j (y, η) = (x, ξ)}, where the canonical transformation χ t,j : T X\0 T X\0 is defined by a flow along the time dependent Hamilton vector field H j = n ( τj τ ) j = ξ j x j x j ξ j in T X\0, from (y, η) at t = 0 to (x, ξ) at t. Treating t as a variable, the canonical relation for the solution operator (4) becomes C = n τ j ξ j x j m {(x, ξ, y, η, t, τ) : τ = τ j (t, ξ), χ t,j (y, η) = (x, ξ)}. Using this representation and asymptotic expansion of D 2 ξξ Φ j with respect to t, it was shown in [16] that under an additional assumption that for each t one of the roots τ j is elliptic, for t outside a discrete set in R, holds rank D 2 ξξ Φ j = n 1. This is for example the case for the wave type equation. For other values of the rank, we have the following Theorem 2.1 Let u(t, x) be the solution of the Cauchy problem (2) and let Φ j (t, x, ξ) solve the eikonal equations (3). Let k = max x,ξ,j rank 2 ξ 2 Φ j(t, x, ξ) (5) 4

5 and assume that k 2. We also write α p = k 1/p 1/2 for 1 < p <. Then if f j L p α+α p j (X), it follows that u(t, ) Lp α(x). Moreover, if k = max x,ξ,j,t [ T,T ] with fixed 0 < T < and k 2, then u(t, ) L p α C T and these orders can not be improved. j=0 rank 2 ξ 2 Φ j(t, x, ξ) f j L p, t [ T, T ], (6) α+αp j In the case of R 1+n with n 4, we do not need any assumptions on k. Theorem 2.2 Let n 4 and let u(t, x) be the solution of the Cauchy problem (2). Let k be defined by (5), not necessarily k 2. Then the conclusions of Theorem 2.1 hold and the orders are sharp. In other function spaces we have Theorem 2.3 In the conditions of Theorems 2.1 and 2.2 the following estimates hold. (i) Let 1 < p q 2. Then u(t, ) L q α C T j=0 f j L p, t [ T, T ], (7) α+αpq j where α pq = n/p+(n k)/q+k/2. The dual result holds for 2 p q <. (ii) For Lipschitz spaces Lip(γ) we have u(t, ) Lip (α) C T j=0 None of the orders above can be improved. f j Lip (α+k/2 j), t [ T, T ]. (8) The best L p properties are exhibited by the operators in (4) for which α p in Theorems 2.1 and 2.2 is zero and there is no loss of smoothness for the solutions. For such operators we can expect k = 0 and according to the representation formula for elliptic operators in [11], they assume the form of a pseudo differential operator composed with a pullback by a smooth coordinate change. However, this turns out to be possible for n = 1 only. 5

6 Theorem 2.4 Let 1 < p <, p 2. Then for every Cauchy data f j L p m j, the solution u(t, ) of (2) belongs to L p m (for small t), if and only if n = 1 and characteristic roots τ j (t, ξ) are linear in ξ. Moreover, in both cases there exist pseudodifferential operators Q jl, S jl Ψ l (Y ) and smooth mappings κ j : X Y such that the solution to (2) is given by u = l=0 m (κ j Q jl )f l = l=0 m (S jl κ j)f l, (9) where κ j are the pullbacks by κ j defined by κ j(f)(x) = f(κ(x)). The Sobolev space estimates (6) hold with α p = 0. Note, that for p = 2 the assumption of this theorem always holds. Remark 2.5 If X is not compact, a local version of Theorems hold for the compactly supported Cauchy data. The solution u(t, ) belongs to the localizations of the corresponding spaces in Theorems Proofs Proof of Theorem 2.1: Let for a fixed ξ the function H j (t, ξ) be the solution of the ordinary differential equation H j t (t, ξ) = τ j(t, ξ), H j (0, ξ) = 0. (10) The symbol σ P is a polynomial in τ, which implies that the roots τ j (t, ξ) are analytic in ξ for ξ 0, and smooth in t in view of the strict hyperbolicity of P. The functions Φ j (t, x, ξ) = x, ξ H j (t, ξ) then satisfy the eikonal equations (3). Because τ j is homogeneous of degree one in ξ, we get that for t sufficiently small, functions H j (t, ξ) are analytic for ξ 0, and homogeneous of degree one in ξ and smooth in t. It follows that the localized Fourier integral operators in (4) are translation invariant and their phase functions are analytic, since we can always cut off the symbol in a neighborhood of ξ = 0 without changing L p estimates, to make the phase function analytic in the support of the symbol. The estimates of Theorem 2.1 now follow from Theorem 7 in [9]. The sharpness follows from the stationary phase argument in Proposition 1 of [11] and the ellipticity of the operators in (4). Proof of Theorem 2.2: This is a direct consequence of Theorem 2.1, where we use Theorem 8 in [9] for the case n = 4, k = 3. Proof of Theorem 2.3: The first part follows from L p estimates of Theorems 2.1 and 2.2 and H 1 L 2 estimates for Fourier integral operators of order n/2 ([19, Ch.3,5.21]). The sharpness argument for any k is as in [11]. The second part is 6

7 the duality argument as in [16], where we use the fact that the translation invariant analytic Fourier integral operators of order k/2 are bounded from H 1 to L 1, which in turn follows from the arguments in Section 5 in [16] and Theorem 2 and Theorem 6 in [9] if k = 1 and k = 2, respectively. Proof of Theorem 2.4: For the completeness of the argument we give the following result, which is Theorem 2 in [11]: Lemma 3.1 Let T I 0 (X, Y ; Λ) be an elliptic Fourier integral operator and assume Λ to be a local graph, 1 < p <, p 2. Then T is continuous from L p comp(y ) to L p loc (X) if and only if there exist P Ψ0 (X), Q Ψ 0 (Y ), such that T = P κ = κ + Q, where κ and κ + are the pullbacks by smooth coordinate changes X Y. We also note that since the canonical relations of κ + and κ viewed as Fourier integral operators, are equal to Λ, it follows that they are the same. Suppose that n 2 and τ j are linear in ξ. Differentiating equation (10) with respect to ξ twice, we get ξξ 2 th j (t, ξ) = 0. Therefore, ξξ 2 H j(t, ξ) = ξξ 2 H j(0, ξ) = 0 and H j is linear in ξ. The rank k in (5) is then equal to zero, which implies the statement because pseudo-differential operators of order j is continuous from (L p α) comp to (L p α+j ). loc Suppose now that P is such that for every Cauchy data f j L p m j, the solution u(t, ) of (2) belongs to L p m. Let T l be an operator of order l in (4). Applying Lemma 3.1 to T = T l (I Δ) l/2 implies formula (9). It also implies that k = 0 and, therefore, ξξ 2 τ j(t, ξ) = t ξξ 2 H j(t, ξ) = 0. It follows that τ j must be a polynomial of degree 1 in ξ. In fact, it is linear in ξ since it also homogeneous of degree one. Now, since P is strictly hyperbolic, all τ j must be different, which is possible only if n = 1. The last statement of Theorem 2.4 on the Sobolev estimates follows from the fact that a pseudo-differential operator of order j is continuous from (L p α) comp to (L p α+j ). loc References [1] M. Beals, L p boundedness of Fourier integrals, Mem. Amer. Math. Soc., 264 (1982). [2] Y. Colin de Verdière, M. Frisch, Régularité Lipschitzienne et solutions de l équation des ondes sur une viriété Riemannienne compacte, Ann. Scient. Ecole Norm. Sup., 9 (1976), [3] J.J. Duistermaat, Fourier integral operators, Birkhäuser, Boston, [4] L. Hörmander, The analysis of linear partial differential operators. Vols. III IV, Springer-Verlag, New York, Berlin, [5] W. Littman, L p L q -estimates for singular integral operators, Proc. Symp. Pure and Appl. Math. A.M.S., 23 (1973) [6] A. Miyachi, On some estimates for the wave operator in L p and H p, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980),

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